b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs
Identifieur interne : 005444 ( Hal/Curation ); précédent : 005443; suivant : 005445b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs
Auteurs : Flavia Bonomo [Argentine] ; Oliver Schaudt [France] ; Maya Stein [Chili] ; Mario Valencia-Pabon [France]Source :
- Algorithmica [ 0178-4617 ] ; 2014.
Abstract
A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by \chi_b(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is called b-continuous if it admits a b-coloring with t colors, for every t = \chi(G),\ldots,\chi_b(G), and b-monotonic if \chi_b(H_1) \geq \chi_b(H_2) for every induced subgraph H_1 of G, and every induced subgraph H_2 of H_1. We investigate the b-chromatic number of graphs with stability number two. These are exactly the complements of triangle-free graphs, thus including all complements of bipartite graphs. The main results of this work are the following: - We characterize the b-colorings of a graph with stability number two in terms of matchings with no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. - We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at most a given threshold. - We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. - Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic.
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DOI: 10.1007/s00453-014-9921-5
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The b-chromatic number of a graph G, denoted by \chi_b(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is called b-continuous if it admits a b-coloring with t colors, for every t = \chi(G),\ldots,\chi_b(G), and b-monotonic if \chi_b(H_1) \geq \chi_b(H_2) for every induced subgraph H_1 of G, and every induced subgraph H_2 of H_1. We investigate the b-chromatic number of graphs with stability number two. These are exactly the complements of triangle-free graphs, thus including all complements of bipartite graphs. The main results of this work are the following: - We characterize the b-colorings of a graph with stability number two in terms of matchings with no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. - We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at most a given threshold. - We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. - Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic.</div></front></TEI><hal api="V3"><titleStmt><title xml:lang="en">b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs</title><author role="aut"><persName><forename type="first">Flavia</forename><surname>Bonomo</surname></persName><email></email><idno type="halauthor">482810</idno><affiliation ref="#struct-92878"></affiliation><affiliation ref="#struct-151993"></affiliation></author><author role="aut"><persName><forename type="first">Oliver</forename><surname>Schaudt</surname></persName><email></email><idno type="halauthor">1115298</idno><affiliation ref="#struct-9459"></affiliation></author><author role="aut"><persName><forename type="first">Maya</forename><surname>Stein</surname></persName><email></email><idno type="halauthor">1115299</idno><affiliation ref="#struct-227891"></affiliation></author><author role="aut"><persName><forename type="first">Mario</forename><surname>Valencia-Pabon</surname></persName><email></email><idno type="halauthor">111523</idno><affiliation ref="#struct-994"></affiliation><affiliation ref="#struct-205125"></affiliation></author><editor role="depositor"><persName><forename>Mario</forename><surname>Valencia</surname></persName><email>valencia@lipn.univ-paris13.fr</email></editor></titleStmt><editionStmt><edition n="v1" type="current"><date type="whenSubmitted">2015-01-12 21:44:06</date><date type="whenModified">2015-09-22 01:12:07</date><date type="whenReleased">2015-01-13 10:06:39</date><date type="whenProduced">2014</date><date type="whenEndEmbargoed">2015-01-12</date><ref type="file" target="https://hal.archives-ouvertes.fr/hal-01102516/document"><date notBefore="2015-01-12"></date></ref><ref type="file" subtype="author" n="1" target="https://hal.archives-ouvertes.fr/hal-01102516/file/b-col-cobip.pdf"><date notBefore="2015-01-12"></date></ref></edition><respStmt><resp>contributor</resp><name key="306281"><persName><forename>Mario</forename><surname>Valencia</surname></persName><email>valencia@lipn.univ-paris13.fr</email></name></respStmt></editionStmt><publicationStmt><distributor>CCSD</distributor><idno type="halId">hal-01102516</idno><idno type="halUri">https://hal.archives-ouvertes.fr/hal-01102516</idno><idno type="halBibtex">bonomo:hal-01102516</idno><idno type="halRefHtml">Algorithmica, Springer Verlag, 2014, pp.17. <10.1007/s00453-014-9921-5></idno><idno type="halRef">Algorithmica, Springer Verlag, 2014, pp.17. <10.1007/s00453-014-9921-5></idno></publicationStmt><seriesStmt><idno type="stamp" n="UNIV-PARIS13">Université Paris-Nord - Paris XIII</idno><idno type="stamp" n="CNRS">CNRS - Centre national de la recherche scientifique</idno><idno type="stamp" n="INRIA">INRIA - Institut National de Recherche en Informatique et en Automatique</idno><idno type="stamp" n="INSMI">CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions</idno><idno type="stamp" n="INRIA-LORRAINE">INRIA Nancy - Grand Est</idno><idno type="stamp" n="LORIA2">Publications du LORIA</idno><idno type="stamp" n="INRIA-NANCY-GRAND-EST">INRIA Nancy - Grand Est</idno><idno type="stamp" n="LORIA-TALC" p="LORIA">Traitement automatique des langues et des connaissances</idno><idno type="stamp" n="LIPN" p="UNIV-PARIS13">Laboratoire d'Informatique de Paris-Nord</idno><idno type="stamp" n="UPMC">Université Pierre et Marie Curie</idno><idno type="stamp" n="UNIV-LORRAINE">Université de Lorraine</idno><idno type="stamp" n="LORIA">LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications</idno><idno type="stamp" n="INRIA_TEST">INRIA - Institut National de Recherche en Informatique et en Automatique</idno></seriesStmt><notesStmt><note type="commentary">An extended abstract of this paper appears in the proceedings of ISCO symposium, LNCS 8596, pp. 100-11, 2014</note><note type="audience" n="2">International</note><note type="popular" n="0">No</note><note type="peer" n="1">Yes</note></notesStmt><sourceDesc><biblStruct><analytic><title xml:lang="en">b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs</title><author role="aut"><persName><forename type="first">Flavia</forename><surname>Bonomo</surname></persName><idno type="halAuthorId">482810</idno><affiliation ref="#struct-92878"></affiliation><affiliation ref="#struct-151993"></affiliation></author><author role="aut"><persName><forename type="first">Oliver</forename><surname>Schaudt</surname></persName><idno type="halAuthorId">1115298</idno><affiliation ref="#struct-9459"></affiliation></author><author role="aut"><persName><forename type="first">Maya</forename><surname>Stein</surname></persName><idno type="halAuthorId">1115299</idno><affiliation ref="#struct-227891"></affiliation></author><author role="aut"><persName><forename type="first">Mario</forename><surname>Valencia-Pabon</surname></persName><idno type="halAuthorId">111523</idno><affiliation ref="#struct-994"></affiliation><affiliation ref="#struct-205125"></affiliation></author></analytic><monogr><idno type="halJournalId" status="VALID">10334</idno><idno type="issn">0178-4617</idno><idno type="eissn">1432-0541</idno><title level="j">Algorithmica</title><imprint><publisher>Springer Verlag</publisher><biblScope unit="pp">17</biblScope><date type="datePub">2014</date></imprint></monogr><idno type="doi">10.1007/s00453-014-9921-5</idno><idno type="arxiv">1310.8313</idno></biblStruct></sourceDesc><profileDesc><langUsage><language ident="en">English</language></langUsage><textClass><classCode scheme="halDomain" n="math">Mathematics [math]</classCode><classCode scheme="halDomain" n="info">Computer Science [cs]</classCode><classCode scheme="halTypology" n="ART">Journal articles</classCode></textClass><abstract xml:lang="en">A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by \chi_b(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is called b-continuous if it admits a b-coloring with t colors, for every t = \chi(G),\ldots,\chi_b(G), and b-monotonic if \chi_b(H_1) \geq \chi_b(H_2) for every induced subgraph H_1 of G, and every induced subgraph H_2 of H_1. We investigate the b-chromatic number of graphs with stability number two. These are exactly the complements of triangle-free graphs, thus including all complements of bipartite graphs. The main results of this work are the following: - We characterize the b-colorings of a graph with stability number two in terms of matchings with no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. - We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at most a given threshold. - We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. - Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic.</abstract></profileDesc></hal></record>Pour manipuler ce document sous Unix (Dilib)
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